After subscribing to a NG, I leave the default most "Don't Synchronize".

Then, whenever I select the NG from the Folders list or I press F5 if the NG
is already selected, this seems to do what I want, namely: update the NG
with headers only for new messages.

What do Synchronize and Catch Up do differently?

Also....

What's the difference between the various Synchronize modes, namely: All
Messages, New Messages Only, and Headers Only?

First, those terms are not mutually exclusive. Does "Headers Only" mean
"All Headers Only", or does it mean "New Message Headers Only"?

Second, when I set "New Messages Only", I believe only headers are
downloaded. I infer that from the fact that when I right-click on a new
message, Properties is grayed-out. I cannot click on Properties until I
double-click the message to read it. (Note: I have deselected Show Preview
Pane.)

Well, the truth is: I do not see any difference between Synchronize New
Messages Only and F5. But that might be because I do not know how/when
Synchronize operates.

Finally....

I believe there are circumstances where OE will delete messages (headers) in
the cache if those messages are no longer on the serve.

Does that always happen? Or does it depend on one or more options that I
select?

If the latter, which options control that behavior? In particular, is it
related to the Synchronize or Catch Up feature?